Limx →-∞[(x^4 sin((1)/(x))+x^2)/((1+|x|^3))]=…………

Name: Problem 32, Chapter 12: 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics
Uploaded: 2021-11-24T17:18:12-08:00
Duration: 1 min 23 s
Channel: Aman Gupta
Description: limx →-∞[(x^4 sin((1)/(x))+x^2)/((1+|x|^3))]=…………

Key Concepts

Dominant Term Analysis

Dominant term analysis is the process of identifying the terms in the numerator and denominator that have the greatest effect on the function's behavior as the independent variable tends to infinity. By comparing the degrees or growth rates of the individual terms, one can simplify the expression to determine the limit, focusing on the highest order terms while neglecting terms which become insignificant in magnitude.

Asymptotic Approximation

Asymptotic approximation involves replacing complicated expressions with simpler ones that approximate their behavior for large values of the variable. In the context of trigonometric functions like sin(1/x), when |x| is large, 1/x becomes small. Therefore, a Taylor series expansion can be used, approximating sin(1/x) by its first term (approximately 1/x), simplifying the analysis of the limit.

Absolute Value Properties

The absolute value function measures the magnitude of a number regardless of its sign. When working with limits that involve |x|, especially as x approaches negative infinity, it is crucial to recognize that |x| is equivalent to -x in that region. This understanding allows for correct simplification of expressions that involve absolute values by converting them into forms that can be more easily analyzed.

Limits at Infinity

This concept deals with evaluating the behavior of a function as the variable approaches positive or negative infinity. It involves understanding how different terms in a function contribute to its overall behavior when the variable becomes very large in magnitude and determining whether the function converges to a specific value, diverges, or oscillates indefinitely.

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